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A239438 Maximal number of points that can be placed on a triangular grid of side n so that there is no pair of adjacent points. 5
1, 1, 3, 4, 6, 7, 10, 12, 15, 19, 22, 26, 31, 35, 40, 46, 51, 57, 64, 70, 77, 85, 92, 100, 109, 117, 126, 136, 145, 155, 166, 176, 187, 199, 210, 222, 235, 247, 260, 274, 287, 301, 316, 330, 345, 361, 376, 392, 409 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

In other words, the independence number of the (n-1)-triangular grid graph.

Apart from a(3) and a(5) same as A007997(n+4) and A058212(n+2). - Eric W. Weisstein, Jun 14 2017

Also the independence number of the n-triangular honeycomb king graph. - Eric W. Weisstein, Sep 06 2017

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

A. V. Geramita, D. Gregory, and L. Roberts, Monomial ideals and points in projective space, J. Pure Applied Alg 40 (1986), pp. 33-62.

Stan Wagon, Graph Theory Problems from Hexagonal and Traditional Chess, The College Mathematics Journal, Vol. 45, No. 4, September 2014, pp. 278-287.

Eric Weisstein's World of Mathematics, Independence Number

Eric Weisstein's World of Mathematics, Triangular Grid Graph

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

FORMULA

a(n) = ceiling(n(n+1)/6) for n > 5, see Geramita, Gregory, & Roberts theorem 5.4. - Charles R Greathouse IV, Dec 04 2014

G.f.: x*(x^9-2*x^8+2*x^7-3*x^6+3*x^5-2*x^4+2*x^3-2*x^2+x-1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Feb 08 2015

EXAMPLE

On a triangular grid of side 5 at most a(5) = 6 points (X) can be placed so that there is no pair of adjacent points.

      X

     . .

    X . X

   . . . .

  X . X . X

MATHEMATICA

Table[1/18 (Piecewise[{{28, n == 2 || n == 4}}, 10] + 3 n (3 + n) + 8 Cos[(2 n Pi)/3]), {n, 0, 20}] (* Eric W. Weisstein, Jun 14 2017 *)

PROG

(PARI) Vec(x*(x^9-2*x^8+2*x^7-3*x^6+3*x^5-2*x^4+2*x^3-2*x^2+x-1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Feb 08 2015

CROSSREFS

Cf. A007997, A058212, A239567.

Sequence in context: A157611 A147609 A202169 * A317138 A032387 A026313

Adjacent sequences:  A239435 A239436 A239437 * A239439 A239440 A239441

KEYWORD

nonn,easy

AUTHOR

Heinrich Ludwig, Mar 18 2014

EXTENSIONS

Extended by Charles R Greathouse IV, Dec 04 2014

STATUS

approved

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Last modified September 15 20:11 EDT 2019. Contains 327086 sequences. (Running on oeis4.)